Heat the oil in a wok or large skillet: Heat 1 tablespoon of vegetable oil in a wok or large skillet over medium-high heat.
Add the chicken: Add the chicken to the wok or skillet and cook until browned and cooked through, about 5-7 minutes. Remove the chicken from the wok or skillet and set aside.
Add the onion and garlic: Add the sliced onion and minced garlic to the wok or skillet and cook until the onion is translucent, about 3-4 minutes.
Add the broccoli: Add the broccoli florets to the wok or skillet and cook until tender, about 3-5 minutes.
Return the chicken to the wok or skillet: Return the chicken to the wok or skillet and stir to combine with the broccoli and onion mixture.
Add the soy sauce, oyster sauce (if using), and sesame oil: Add the soy sauce, oyster sauce (if using), and sesame oil to the wok or skillet and stir to combine.
Season with salt and pepper: Season the chicken and broccoli mixture with salt and pepper to taste.
Serve: Serve the chicken and broccoli stir-fry hot, garnished with chopped green onions and sesame seeds (if desired).